Official Answer

Direct attempts to solve for x in this problem will run into quadratics that don't factor and horrible non-integers that need to be raised to fourth powers. Instead, let's focus on manipulating the equation to solve for x^4 + x^−4 directly.

Given the similar structure of the given information, it seems reasonable to begin by squaring the equation x^1+x^−1=5. Be careful, though, not to simply square each term; exponents do not distribute over addition. Instead, recognize the special quadratic. We're looking at two terms added and then squared, so this expression fits the form (a+b)^2=a^2+2ab+b^2. Thus our result will be

(x^1+x^−1^2=5^2

(x^1)^2+2(x^1)(x^−1)+(x^−1)^2=25

x^2+2+x^−2=25

x^2+x^−2=23

Now just repeat the process of squaring both sides once more:

(x^2+x^−2)^2=23^2

x^4+2(x^2)(x^−2)+x^−4=23^2

x^4+2+x^−4=23^2

x^4+x^−4=23^2−2

And it's not even really necessary to calculate 23^2 (which turns out to be 529). 23^2 must end in a 9, so 23^2−2 must end in a 7, and the answer has to be A.

_________________

Kudos if you like my response please